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Cohomological field theories and generalized Seiberg-Witten equations

Shuhan Jiang, Jürgen Jost

TL;DR

The paper develops a universal framework to construct cohomological field theories from nonlinear PDEs, with a detailed application to the generalized Seiberg–Witten system that unifies the Donaldson–Witten, Seiberg–Witten, and Kapustin–Witten functionals. It casts CohFTs in a derived-geometric setting, distinguishing a minimal construction from a standard extension that captures non-free gauge actions, and demonstrates connections to the Mathai–Quillen formalism and Atiyah–Jeffrey localization. Observables arise via an equivariant de Rham perspective on solution spaces, and a BV-based quantization program is proposed to define manifold invariants and quantum cohomologies. The framework also suggests broader applicability to sigma-model-type theories and potential super-extensions, outlining future mathematical and physical developments in quantization and invariants for 4-manifolds.

Abstract

We introduce a formalism for constructing cohomological field theories (CohFT) out of nonlinear PDEs based on the first author's previous work (arXiv:2202.12425). We apply the formalism to the generalized Seiberg-Witten equations and show that the obtained CohFT functionals agree with the existing ones proposed by physicists. This leads to a unified perspective from which to view the full supersymmetric functionals of the Donaldson-Witten, Seiberg-Witten, and Kapustin-Witten theories and understand the relations between them. We also outline a quantization program for our framework and discuss its potential to produce manifold invariants and quantum cohomologies.

Cohomological field theories and generalized Seiberg-Witten equations

TL;DR

The paper develops a universal framework to construct cohomological field theories from nonlinear PDEs, with a detailed application to the generalized Seiberg–Witten system that unifies the Donaldson–Witten, Seiberg–Witten, and Kapustin–Witten functionals. It casts CohFTs in a derived-geometric setting, distinguishing a minimal construction from a standard extension that captures non-free gauge actions, and demonstrates connections to the Mathai–Quillen formalism and Atiyah–Jeffrey localization. Observables arise via an equivariant de Rham perspective on solution spaces, and a BV-based quantization program is proposed to define manifold invariants and quantum cohomologies. The framework also suggests broader applicability to sigma-model-type theories and potential super-extensions, outlining future mathematical and physical developments in quantization and invariants for 4-manifolds.

Abstract

We introduce a formalism for constructing cohomological field theories (CohFT) out of nonlinear PDEs based on the first author's previous work (arXiv:2202.12425). We apply the formalism to the generalized Seiberg-Witten equations and show that the obtained CohFT functionals agree with the existing ones proposed by physicists. This leads to a unified perspective from which to view the full supersymmetric functionals of the Donaldson-Witten, Seiberg-Witten, and Kapustin-Witten theories and understand the relations between them. We also outline a quantization program for our framework and discuss its potential to produce manifold invariants and quantum cohomologies.
Paper Structure (15 sections, 7 theorems, 139 equations, 2 figures, 1 table)

This paper contains 15 sections, 7 theorems, 139 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The general construction
  3. The minimal construction of CohFTs
  4. The standard construction of CohFTs
  5. The 0-dimensional toy model
  6. Mathai--Quillen formalism
  7. The Atiyah--Jeffrey formula
  8. Generalized Seiberg--Witten theory
  9. Donaldson--Witten theory
  10. Seiberg--Witten theory
  11. Kapustin--Witten theory
  12. Symmetries and observables
  13. Vector supersymmetries and observables of the generalized Seiberg--Witten theory
  14. Quantization program
  15. Conclusions and discussions

Key Result

Lemma 2.1

$\rho: \mathfrak{g}_{dR} \rightarrow \mathfrak{X}(T[1]\mathcal{M})$ defines a $\mathfrak{g}_{dR}$-module structure on $C^{\infty}(T[1]\mathcal{M})$.

Figures (2)

  • Figure 2.1: A picture of $\mathrm{pr}^{-1}(\mathrm{Sol}(\mathcal{F})_{red}) \subset \widetilde{\mathrm{Sol}(\mathcal{F})}$.
  • Figure 6.1: A picture about how to define the quantum product on $\mathbf{Obs}^{\mathcal{F}}_{cl}(B_r(0))$.

Theorems & Definitions (29)

  • Lemma 2.1
  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • proof
  • Definition 2.1
  • Remark 2.3
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 19 more